Biography:Édouard Goursat
Édouard Goursat | |
|---|---|
 Edouard Goursat | |
| Born | 21 May 1858 Lanzac, Lot |
| Died | 25 November 1936 (aged 78) |
| Nationality | France |
| Alma mater | École Normale Supérieure |
| Known for | Goursat tetrahedron Goursat theorem Goursat's lemma Inverse function theorem |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of Paris |
| Doctoral advisor | Jean Gaston Darboux |
| Doctoral students | Georges Darmois Dumitru Ionescu |
Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century. It set a standard for the high-level teaching of mathematical analysis, especially complex analysis. This text was reviewed by William Fogg Osgood for the Bulletin of the American Mathematical Society.[1][2] This led to its translation into English by Earle Raymond Hedrick published by Ginn and Company. Goursat also published texts on partial differential equations and hypergeometric series.
Life
Edouard Goursat was born in Lanzac, Lot. He was a graduate of the École Normale Supérieure, where he later taught and developed his Cours. At that time the topological foundations of complex analysis were still not clarified, with the Jordan curve theorem considered a challenge to mathematical rigour (as it would remain until L. E. J. Brouwer took in hand the approach from combinatorial topology). Goursat's work was considered by his contemporaries, including G. H. Hardy, to be exemplary in facing up to the difficulties inherent in stating the fundamental Cauchy integral theorem properly. For that reason it is sometimes called the Cauchy–Goursat theorem.
Work
Goursat, along with Möbius, Schläfli, Cayley, Riemann, Clifford and others, was one of the 19th century mathematicians who envisioned and explored a geometry of more than three dimensions.[3]
He was the first to enumerate the finite groups generated by reflections in four-dimensional space, in 1889.[4] The Goursat tetrahedra are the fundamental domains which generate, by repeated reflections of their faces, uniform polyhedra and their honeycombs which fill three-dimensional space. Goursat recognized that the honeycombs are four-dimensional Euclidean polytopes.
He derived a formula for the general displacement in four dimensions preserving the origin, which he recognized as a double rotation in two completely orthogonal planes.[5]
Goursat was the first to note that the generalized Stokes theorem can be written in the simple form
- [math]\displaystyle{ \int_S \omega = \int_T d \omega }[/math]
where [math]\displaystyle{ \omega }[/math] is a p-form in n-space and S is the p-dimensional boundary of the (p + 1)-dimensional region T. Goursat also used differential forms to state the Poincaré lemma and its converse, namely, that if [math]\displaystyle{ \omega }[/math] is a p-form, then [math]\displaystyle{ d\omega=0 }[/math] if and only if there is a (p − 1)-form [math]\displaystyle{ \eta }[/math] with [math]\displaystyle{ d \eta=\omega }[/math]. However Goursat did not notice that the "only if" part of the result depends on the domain of [math]\displaystyle{ \omega }[/math] and is not true in general. Élie Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development of the De Rham cohomology of a differential manifold.
Books by Edouard Goursat
- A Course In Mathematical Analysis Vol I Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1904)
- A Course In Mathematical Analysis Vol II, part I Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1916) (Complex analysis)
- A Course In Mathematical Analysis Vol II Part II Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1917) (Differential Equations)
- Leçons sur l'intégration des équations aux dérivées partielles du premier ordre (Hermann, Paris, 1891)[6]
- Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 1[yes|permanent dead link|dead link}}] (Hermann, Paris 1896–1898)[6]
- Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 2[yes|permanent dead link|dead link}}] (Hermann, Paris 1896–1898)[6]
- Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent[yes|permanent dead link|dead link}}] (Hermann, Paris, 1936–1939)[7]
- Le problème de Bäcklund[yes|permanent dead link|dead link}}] (Gauthier-Villars, Paris, 1925)
- Leçons sur le problème de Pfaff[yes|permanent dead link|dead link}}] (Hermann, Paris, 1922)[8]
- Théorie des fonctions algébriques et de leurs intégrales : étude des fonctions analytiques sur une surface de Riemann[yes|permanent dead link|dead link}}] with Paul Appell (Gauthier-Villars, Paris, 1895)[9]
- Théorie des fonctions algébriques d'une variable et des transcendantes qui s'y rattachent Tome II, Fonctions automorphes[yes|permanent dead link|dead link}}] with Paul Appell (Gauthier-Villars, 1930)
See also
- Goursat problem
- Goursat structure
- Goursat's lemma
References
- ↑ Osgood, W. F. (1903). "Review: Cours d'analyse mathématique. Tome I.". Bull. Amer. Math. Soc. 9 (10): 547–555. doi:10.1090/s0002-9904-1903-01028-3. http://www.ams.org/journals/bull/1903-09-10/S0002-9904-1903-01028-3/.
- ↑ Osgood, W. F. (1908). "Review: Cours d'analyse mathématique. Tome II.". Bull. Amer. Math. Soc. 15 (3): 120–126. doi:10.1090/s0002-9904-1908-01704-x. http://www.ams.org/journals/bull/1908-15-03/S0002-9904-1908-01704-X/.
- ↑ Stillwell, John (January 2001). "The Story of the 120-Cell". Notices of the AMS 48 (1): 17-25. https://www.ams.org/notices/200101/fea-stillwell.pdf.
- ↑ Coxeter 1973, p. 209, §11.x.
- ↑ Coxeter 1973, p. 216, §12.1 Orthogonal transformations.
- ↑ 6.0 6.1 6.2 Lovett, Edgar Odell (1898). "Review: Goursat's Partial Differential Equations". Bull. Amer. Math. Soc. 4 (9): 452–487. doi:10.1090/S0002-9904-1898-00540-2. http://www.ams.org/journals/bull/1898-04-09/S0002-9904-1898-00540-2/.
- ↑ Szegő, G. (1938). "Review: Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent by É. Goursat". Bull. Amer. Math. Soc. 44 (1, Part 1): 16–17. doi:10.1090/s0002-9904-1938-06652-9. http://www.ams.org/journals/bull/1938-44-01/S0002-9904-1938-06654-2/S0002-9904-1938-06654-2.pdf.
- ↑ Dresden, Arnold (1924). "Review: Leçons sur le problème de Pfaff". Bull. Amer. Math. Soc. 30 (7): 359–362. doi:10.1090/s0002-9904-1924-03903-2. http://www.ams.org/journals/bull/1924-30-07/S0002-9904-1924-03903-2/.
- ↑ Osgood, W. F. (1896). "Review: Théorie des fonctions algébriques et de leurs intégrales, by P. Appell and É. Goursat". Bull. Amer. Math. Soc. 2 (10): 317–327. doi:10.1090/s0002-9904-1896-00353-0. http://www.ams.org/journals/bull/1896-02-10/S0002-9904-1896-00353-0/.
- Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
- Katz, Victor (2009). A History of Mathematics: An introduction (3rd ed.). Boston: Addison-Wesley. ISBN 978-0-321-38700-4.
External links
- O'Connor, John J.; Robertson, Edmund F., "Édouard Goursat", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Goursat.html.
- William Fogg Osgood A modern French Calculus Bull. Amer. Math. Soc. 9, (1903), pp. 547–555.
- William Fogg Osgood Review: Edouard Goursat, A Course in Mathematical Analysis Bull. Amer. Math. Soc. 12, (1906), p. 263.
- Édouard Goursat at the Mathematics Genealogy Project
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